Btw I'd be happy to talk with WebHubbleTelescope on the issue.

He at least seems to understands the problem.
He rejected HL and generally I AGREE WITH HIM.
Sorry for the caps but your not listening.
Someone has already done a fantastic job of questioning HL
and he used the correct production profile.

Until you integrate his work I'm not sure what the heck your doing.

http://mobjectivist.blogspot.com/2005/12/hubbert-linearization.html

One more quote from his blog.

I checked the math on this, and it really gets you thinking about what data visualization expert Tufte says about graphing data in a biased fashion. That convergence on a continuously shrinking error acts like a laser beam and gives people the impression of an excellent fit that may have dubious value at best.

So a good well reasoned rebuke of HL already exists.
And I'll say one more time generally I agree with him.
But its not clear that he can come up with a better model given the data we have. Not that he can't create a better model or a better model is possible simply do we have enough
data to support a better model.

So again does HL have problems yes here is the link that points out its flaws.

My answer as to why it works is simple.

Although the function changes that describes the actual production profile HL implicitly assumes that the rate of production is related to the overall URR via the logistic curve. Since we know from theoretical plate models that the time of elution or peak is related to the interaction of the
material with environment with a given geology if you steadily pump a field the time of peak does not change.
This means you can change the shape of the curve but your not actually able to change URR or the date of peak by much
without massive changes in the way the field is pumped.

In the case of chromatography they use Guassians and derive the interaction numbers i.e theoretical plates.
Now using a model that is well understood and tested the Theoretical Plate model and applying it to a oil field its says the following. If I drill a well into a porous geologic formation and a few wells around it. And first I pump some oil down it then start pumping water. The wells in a circle around the pumped well will get the oil in a Gaussian profile. The main body of the oil has a interaction with its it surroundings thats FIXED!
In the case of a field full of oil this block of oil is moving through a system that has immobile oil as part of its
environment but the behavior is no different. As you begin to produce a field the main driving force is oil pushing oil. Later its water pushing oil but the little Gaussian regions can't move till the ones behind them move.

This is my interpretation of what Fractional_Flow says and he
is also correct its the field geology that determines the peak.
http://europe.theoildrum.com/node/2372#comment-170481

Hubbert chose the logistic and uses the rate to guess the URR. The choice of logistic is interesting and its not clear
in the least its the best and again its not clear that a better one exists given the data we have. Given that we are just doing a taylor expansion the exact shape of the curve are not important whats important or interesting is that HL works when the production is assumed to be a curve.
The examples you have given don't even behave correctly to the first taylor expansion term no wonder they blow up.

I'm only saying you need to use a production profile that can be taylor series expanded about its center point i.e it needs to look like a parabola to apply HL otherwise its junk. I don't need to do anything the work is already done and has been done for some time. You simply need to use a production profile thats reasonably close to what HL assumes.

Next since your generating data if you pick parabolas which are simple you can find one that gives a perfect match then vary the parabolas away from the perfect keeping the area under the curve or URR constant. one to see how much HL varies. Actually you can use any series of parameterized curves. The only restriction is they all have to have the same URR.

Your current work is not even close to being the right way to critique HL.